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\section{\label{sec:DUFF}The DUFF Force Field} |
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\section{The DUFF Energy Functionals} |
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The DUFF (\underline{D}ipolar \underline{U}nified-atom |
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\underline{F}orce \underline{F}ield) force field was developed to |
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simulate lipid bilayer formation and equilibrium dynamics. We needed a |
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model capable of forming bilaers, while still being sufficiently |
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computationally efficient allowing simulations of large systems |
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(\~100's of phospholipids, \~1000's of waters) for long times (\~10's |
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of nanoseconds). |
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The main energy functional set in OOPSE is DUFF (the Dipolar |
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Unified-atom Force Field). DUFF is a collection of parameters taken |
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from Seipman \emph{et al.}\cite{Siepmann1998}~and Ichiye \emph{et |
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With this goal in mind, we decided to eliminate all charged |
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interactions within the force field. Charge distributions were |
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replaced with dipolar entities, and charge neutral distributions were |
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reduced to Lennard-Jones interaction sites. This simplification cuts |
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the length scale of long range interactions from $\frac{1}{r}$ to |
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$\frac{1}{r^3}$ (Eq.~\ref{eq:dipole} vs.~ Eq.~\ref{eq:coloumb}), |
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allowing us to avoid the computationally expensive Ewald-Sum. Instead, |
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we can use neighbor-lists and cutoff radii for the dipolar |
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interactions. |
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\begin{align} |
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V^{\text{dipole}}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
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\boldsymbol{\Omega}_{j}) &= |
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\frac{1}{4\pi\epsilon_{0}} \biggl[ |
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\frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}} |
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- |
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\frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) % |
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(\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) } |
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{r^{5}_{ij}} \biggr]\label{eq:dipole} \\ |
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V^{\text{ch}}_{ij}(\mathbf{r}_{ij}) &= \frac{q_{i}q_{j}}% |
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{4\pi\epsilon_{0} r_{ij}} \label{eq:coloumb} |
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\end{align} |
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Applying this standard to the lipid model, we decided to represent the |
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lipid model as a point dipole interaction site. Lipid head groups are |
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typically zwitterionic in nature, with sometimes full integer charges |
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seperated by only 5 to 6~$\mbox{\AA}$. By placing a dipole of |
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20.6~Debye at the head groups center of mass, our model mimics the |
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dipole of DMPC.\cite{Cevc87} Then, to account for the steric henderanc |
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of the head group, a Lennard-Jones interaction site is also oacted at |
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the psuedoatom's center of mass. The model is illustrated in |
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Fig.~\ref{fig:lipidModel}. |
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\begin{figure} |
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\includegraphics[angle=-90,width=\linewidth]{lipidModel.epsi} |
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\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ % |
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is the bend angle, $\mu$ is the dipole moment of the head group, and n is the chain length.} |
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\label{fig:lipidModel} |
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\end{figure} |
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Turning to the tail chains of the phospholipids, we needed a set of |
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scalable parameters to model the alkyl groups as Lennard-Jones |
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interaction sites. For this, we used the TraPPE force field of |
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Siepmann \emph{et al}.\cite{Siepmann1998} The force field is a |
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unified-atom representation of n-alkanes. It is parametrized against |
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phase equilibria using Gibbs Monte Carlo simulation techniques. One of |
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the advantages of TraPPE is that is generalizes the types of atoms in |
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an alkyl chain to keep the number of pseudoatoms to a minimum. |
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%( $ \mbox{CH_3} $ %-$\mathbf{\mbox{CH_2}}$-$\mbox{CH_3}$ is the same as |
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Another advantage of using TraPPE is the constraining of all bonds to |
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be of fixed length. Typically, bond vibrations are the motions in a |
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molecular dynamic simulation. This neccesitates a small time step |
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between force evaluations be used to ensure adequate sampling of the |
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bond potential. Failure to do so will result in loss of energy |
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conservation within the microcanonical ensemble. By constraining this |
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degree of freedom, time steps larger than were previously allowable |
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are able to be used when integrating the equations of motion. |
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The main energy function in OOPSE is DUFF (the Dipolar Unified-atom |
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Force Field). DUFF is a collection of parameters taken from Seipmann |
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and Ichiye \emph{et |
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al.}\cite{liu96:new_model} The total energy of interaction is given by |
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Eq.~\ref{eq:totalPotential}. |
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Eq.~\ref{eq:totalPotential}: |
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\begin{equation} |
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V_{\text{lipid}} = |
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\sum_{i}V_{i}^{\text{internal}} |
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+ \sum_i \sum_{j>i} \sum_{\alpha_i} |
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\sum_{\beta_j} |
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V_{\text{LJ}}(r_{\alpha_{i}\beta_{j}}) |
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+\sum_i\sum_{j>i}V_{\text{dp}}(r_{1_i,1_j},\Omega_{1_i},\Omega_{1_j}) |
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\label{eq:totalPotential} |
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V_{\text{Total}} = |
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\overbrace{V_{\theta} + V_{\phi}}^{\text{bonded}} + |
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\underbrace{V_{\text{LJ}} + V_{\text{Dp}} + % |
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V_{\text{SSD}}}_{\text{non-bonded}} \label{eq:totalPotential} |
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\end{equation} |
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\subsection{Bonded Interactions} |
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\label{subSec:bondedInteractions} |
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The bonded interactions in the DUFF functional set are limited to the |
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bend potential and the torsional potential. Bond potentials are not |
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calculated, instead all bond lengths are fixed to allow for large time |
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steps to be taken between force evaluations. |
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The bend functional is of the form: |
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\begin{equation} |
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V_{\theta} = \sum k_{\theta}( \theta - \theta_0 )^2 \label{eq:bendPot} |
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\end{equation} |
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$k_{\theta}$, the force constant, and $\theta_0$, the equilibrium bend |
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angle, were taken from the TraPPE forcefield of Siepmann. |
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The torsion functional has the form: |
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\begin{equation} |
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V_{\phi} = \sum ( k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0) |
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\label{eq:torsionPot} |
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\end{equation} |
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Here, the authors decided to use a potential in terms of a power |
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expansion in $\cos \phi$ rather than the typical expansion in |
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$\phi$. This prevents the need for repeated trigonemtric |
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evaluations. Again, all $k_n$ constants were based on those in TraPPE. |
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\subsection{Non-Bonded Interactions} |
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\label{subSec:nonBondedInteractions} |
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\begin{equation} |
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V_{\text{LJ}} = \text{internal + external} |
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\end{equation} |
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